There are hobbyists claiming up to 300x (at this level you can clearly see bacteria), by stacking two lenses.
In any case, there are extremely cheap DIY solutions out there, but it may not be worth it for you.
Some ideas: https://www.wired.com/2011/03/diy-cellphone-microscope/
If the son has any real interest in microscopy, I think the money for a proper one is well invested.
For 10x magnification, good for looking at bugs, plants, and money, don't get a microscope - get a loupe (the thing jewelers use). Get a glass one with an LED light. A good one will be about $60.
For 100x magnification, spend around $300. Mine is an Amscope and it works great. Again make sure you get one with an LED light. Get one with the light above the platform unless preparing slides sounds fun.
If you want to look at any cellular structures, I don't think there's a way around slides and bottom-up lighting. Fortunately, preparing slides is actually fun ;-) Also, all serious microscopy I know is done like that (unless you're using stereo microscopes for entomology), so one might as well learn how to do it properly.
The former had a rubbish microscope, but a really neat box with plenty of accessories. The second was, like I said, ancient (one of those old brass microscopes with a mirror for lighting!), but with very good optics. With a few more odds and ends, they formed my little "home lab".
One of the things that got me started on my career path to biology...
There were a few sources of frustration:
1. The built in lamp was way too weak, but there was also a mirror, but positioning the microscope to get light from the window was a pain. The biggest difference to the awesome school microscopes was that they had great light.
2. The microscope only came with 5 or so glass plates to put your samples on. I could only look at a handful of samples, and had to wash my glass plates again, etc. Until one day I just "borrowed" a small box of glass plates from school (I guess I should have just asked my parents for more plates, they don't cost much, but you need lots of them)
3) My microscope didn't have any mechanics to move the sample, which made it pretty tricky to move especially at higher magnifications
4) I also got a box of prepared tissue samples with the microscope, but didn't really understand what to do with them. They had nice colors, though.
We found some squash bug eggs in the garden and put these on a slide, it was a good activity.
What I can strongly dis-recommend is to buy a set from Barnes & Noble. I bought a microscope kit that came in some sort of plastic "suitcase". It had the microscope itself and all sorts of accessories, like prepared slides. The image quality was awful.
In the "Related methods" section of that article, you can see several things that look like nice modern upgrades that are probably expensive.
They're so small that you can always have them with you to look at specimens on the spot, especially useful during walks in nature.
Bought several of them as gifts as well, can recommend!
It has a plug in LED light and also a mirror. You need to prepare slides of whatever you're looking at, as the light comes from below.